Multifractals and critical phenomena in percolating networks: Fixed point, gap scaling, and universality

Fourcade, B.; Breton, P.; Tremblay, A.–M. S.

doi:10.1103/physrevb.36.8925

Cited by 30 publications

(7 citation statements)

References 24 publications

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“…monofractals with a single dimension and multifractals with a series of dimensions, are employed as model objects in studying diverse physical problems, e.g. in turbulent flow [8,9], morphology of fracture surfaces [10][11][12], porous structure of solids [13][14][15], theory of percolation [16][17][18], Brownian motion [1][2][3]6], statistics of polymer strings [19], colloid aggregates [20,21] and theory of dielectric breakdown [22][23][24][25], to mention just some of them. In other words, fractals have become a useful physical tool able to model diverse features of physical systems.…”

Section: Introductionmentioning

confidence: 99%

Deterministic fractals

Ficker¹,

Benesovský²

2002

Eur. J. Phys.

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Section: Introductionmentioning

confidence: 99%

Deterministic fractals

Ficker¹,

Benesovský²

2002

Eur. J. Phys.

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“…In the absence of symmetry breaking in replica space, the modulus of the replicated k in Eq. ( 21) is equal to k in expression (15).…”

Section: B Replica Methods and Effective Actionmentioning

confidence: 99%

“…Finally, let us recall that the positive integer moments, < α i 2n α > C , suffice to characterize completely the distribution of the currents flowing through the network. [15] But, as stated previously, the exact value of the integer moments is necessary to reconstruct all the information so that the leading scaling behavior of the positive moments does not suffice to find, for example, negative moments. In the following, we concentrate only on the scaling behavior of the positive integer moments.…”

Section: Phenomenology Of Multifractals In Percolationmentioning

confidence: 99%

Field theory and second renormalization group for multifractals in percolation

Fourcade

Tremblay

1995

Phys. Rev. E

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The field-theory for multifractals in percolation is reformulated in such a way that multifractal exponents clearly appear as eigenvalues of a second renormalization group. The first renormalization group describes geometrical properties of percolation clusters, while the second-one describes electrical properties, including noise cumulants. In this context, multifractal exponents are associated with symmetry-breaking fields in replica space. This provides an explanation for their observability. It is suggested that multifractal exponents are "dominant" instead of "relevant" since there exists an arbitrary scale factor which can change their sign from positive to negative without changing the Physics of the problem.

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“…The right-hand side therefore gives a rigorous lower boundary for χ e (β), although not a directly computable bound since g e is not known exactly. Equation ( 11) also corresponds in a sense to the assumption of 'gap scaling' which is precisely what does not occur for multifractal exponents [27][28][29][30].…”

Section: Effective-medium Approximationmentioning

confidence: 99%

Critical behaviour of non-linear susceptibility in random non-linear resistor networks

Zhang¹

1996

J. Phys.: Condens. Matter

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The critical behaviour of non-linear susceptibility of a two-component composite is studied in this paper. The first component of fraction p is non-linear and obeys a current-field (J -E) characteristic of the form J = g 1 v + χ 1 v β while the second component of fraction q is linear with J = g 2 v. Near the percolation threshold p c or q c , we examine the conductorinsulator (C-I) limit (g 2 = 0) and superconductor-conductor (S-C) limit (g 2 = +∞). For the C-I limit and p > p c , the effective linear and non-linear response functions behave as g e ≈ (p − p c ) t and χ e (β) ≈ (p − p c ) t 2 (β) , respectively. For the S-C limit and q < q c , g e and χ e (β) are found to diverge as g e ≈ (q c − q) −s and χ e (β) ≈ (q c − q) s 2 (β) . Within the effectivemedium approximation, the exponents are found to be s = t = 1 and s 2 (β) = t 2 (β) = (β +1)/2, p c = 1/d and q c = (d − 1)/d. By using a connection between the non-linear response of the random non-linear composite problem and the resistance or conductance fluctuations of the corresponding random linear composite problem, the exponents t 2 (β) and s 2 (β) are found to berespectively, where t (s) is the conductivity exponent in a C-I(S-C) composite, d is the dimension of the composite and ν is the correlation-length exponent in d dimensions, κ((β + 1)/2) and κ ((β + 1)/2) are given by ψ R ((β + 1)/2) + [(β + 1)/2](dν − ζ R ) = κ((β + 1)/2) + dν, ψ G ((β + 1)/2) + [(β + 1)/2](dν − ζ G ) = κ ((β + 1)/2) + dν, where ψ R ((β + 1)/2)(ψ G ((β + 1)/2)) characterizes the scaling of the [(β + 1)/2]th cumulant of the global resistance (conductance) distribution due to local resistance (conductance) fluctuations in the corresponding linear C-I(S-C) composites, ζ R = t − (d − 2)ν and ζ G = s + (d − 2)ν.We prove that t 2 (β) is a monotonically increasing function of β while s 2 (β) is a monotonically decreasing function of β, which have the following special values:The critical behaviour of the non-linear susceptibility in a C-I composite is very different from that of the non-linear susceptibility in a S-C composite and some unexpected results of the critical behaviour of non-linear susceptibility in a C-I network are reported in this paper.

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Multifractals and critical phenomena in percolating networks: Fixed point, gap scaling, and universality

Cited by 30 publications

References 24 publications

Deterministic fractals

Deterministic fractals

Field theory and second renormalization group for multifractals in percolation

Critical behaviour of non-linear susceptibility in random non-linear resistor networks

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